Point group

point group is a group of symmetry operations all of which leave at least one point unmoved. A crystallographic point group is a point group that  maps  a point lattice onto itself: in three dimensions rotations and rotoinversions are  restricted  to 1, 2, 3, 4, 6 and $\bar 1$$\bar 2$ (= m), $\bar 3$$\bar 4$$\bar 6$ respectively.

Occurrence

Crystallographic point groups occur:

• in vector space, as symmetries of the external shapes of crystals (morphological symmetry), as well as symmetry of the physical properties of the crystal ("vector point group");
• in point space, as site-symmetry groups of points in lattices or in crystal structures, and as symmetries of atomic groups and coordination polyedra ("point point group").